function," estimating the number of primes. but on the other hand, when we step back, then what happens? >> right, well, then the curving was much more continuous. in fact, it converges to a very exact and beautiful curve given by the natural logarithm function. >> and gauss -- gauss predicted this, but in fact he didn't prove it. >> no, that had to wait until about 200 years later, i think, to be proven. >> so this very smooth curve is now sort of canonized in what's called the "prime number theorem," but in fact we now know today that it doesn't quite go far enough. it's an estimate, but you could do better, and doing better was what was in the mind of a student who was at the same university where gauss was teaching, bernhard riemann. is that right? >> yes, yes. so riemann studied the area between the jagged function, which counts the primes, and this smooth curve. >> that's right, so the smooth curve again is some approximation, but the actual number, it's not getting right generally. >> right, it's either above -- overshoots or undershoots, and so there's this funny area that you have to understand.